U.S. President Donald Trump has said the United States is "not getting along with Russia at all" and that bilateral relations may be at an "all-time low."

Trump made the comments at the White House on April 12, after Secretary of State Rex Tillerson met with Russian President Vladimir Putin in Moscow -- the highest-level face-to-face talks between the two sides under President Donald Trump's administration.

"Things went pretty well," he said during a news conference with NATO Secretary-General Jens Soltenberg. "Maybe better than anticipated."

Trump expressed hope that he can improve relations with Putin, adding, "We're going to see what happens."

"It would be a fantastic thing" if the two world powers got along better, said Trump, who cautioned that "it may be just the opposite."

Trump's comments came amid heightened tensions over Moscow’s backing of Syrian President Bashar al-Assad following a suspected chemical weapons attack in Syria last week.

Damascus and Moscow claim the toxic gas that killed dozens of civilians in a rebel-held Syrian town on April 4 was released when government bombs struck a rebel-controlled chemical-weapons depot -- an assertion that Western governments reject.

Washington responded to the suspected chemical attack by targeting a Syrian air base with a barrage of cruise missiles in an operation that incensed Moscow.

Following the meeting with Putin and earlier talks with Foreign Minister Sergei Lavrov in Moscow, Tillerson told reporters that bilateral ties are "at a low point."

"The world's two foremost nuclear powers cannot have this kind of relationship," he said at a news conference alongside Lavrov.

The Russian foreign minister said that "despite the quantity of existing problems... there are considerable prospects for joint work."

He added that he and Tillerson had agreed to designate representatives on each side specifically tasked with addressing "irritants" in bilateral relations.

Tillerson said the two countries had agreed to set up a "working group" to coordinate on "smaller issues." Such collaboration, he said, could help the two sides later address "more serious problems."

However, there was no indication that the two sides changed their positions on the issue of Syria, which triggered a spate of mutual rebukes in the run-up to -- and during -- Tillerson's visit.

At the White House press conference, Trump, who has repeatedly questioned NATO’s purpose, also said the Western military alliance is "no longer obsolete."

He hailed NATO’s role in the fight against terrorism, describing the alliance as the "bulwark of international peace and security."

Trump repeated his call for other member states to contribute more funding to the alliance, saying European states "must pay what they owe."

He also called on the alliance to do more to help Iraqi and Afghan "partners."